Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is :

### Concept & Logic When trains cross a stationary man, the time taken is proportional to their lengths relative to their own speeds ($L = v \times t$). When crossing each other, the total time depends on the sum of their lengths divided by their relative speeds. We can use alligation to find the ratio of their speeds directly. $$ \text{Speed Ratio} = \frac{t_{mix} - t_2}{t_1 - t_{mix}} $$ ### Step-by-Step Solution 1. **Traditional Setup:** - Let speeds be $v_1$ and $v_2$. - Length of train 1, $L_1 = 27 v_1$. - Length of train 2, $L_2 = 17 v_2$. 2. **Crossing Each Other:** - Time to cross each other = $23$ seconds. - $\frac{L_1 + L_2}{v_1 + v_2} = 23$. - Substitute the lengths: $\frac{27 v_1 + 17 v_2}{v_1 + v_2} = 23$. 3. **Solve for Ratio:** - $27 v_1 + 17 v_2 = 23 v_1 + 23 v_2$. - $27 v_1 - 23 v_1 = 23 v_2 - 17 v_2$. - $4 v_1 = 6 v_2$. - $\frac{v_1}{v_2} = \frac{6}{4} = \frac{3}{2}$. ### Exam Strategy & Shortcut Use the **Rule of Alligation** for a lightning-fast solution. - Time 1 = 27 - Time 2 = 17 - Mean Time = 23 Ratio of speeds = $(23 - 17) : (27 - 23) = 6 : 4 = 3 : 2$. ### Common Pitfall A common mistake is trying to calculate the lengths of the trains without recognizing that the speeds and lengths are interdependent, resulting in a perceived lack of data. ### Final Answer Therefore, the correct answer is **3 : 2**.